Generalizing Green's Syzygy Theorems to Symmetric Products of Curves

Abstract

Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on $n$, using a filtration of the kernel bundle of $T_{n,L}$ to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when $g\geq1$ and $n\geq2$, $T_{n,L}$ satisfies property $(N_p)$ for $\mathrm{deg}(L)\geq(n+1)g+p$. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.